Question

Using 5 conveyances, the number of ways of making 3 journeys is?
A. $3 \times 5$
B. ${3^5}$
C. ${5^3}$
D. ${5^3} - 1$

Answer 1

Hint: We have 5 conveyances and we need to make 3 journeys. So, we can take one of the 5 conveyances for making the 1st journey. Similarly, for the 2nd journey also we can use any of the 5 conveyances and for the 3rd journey also we can use any of the 5 conveyances. Then we can find the total number of ways of making 3 journeys by taking their product.

Complete step-by-step answer:
We have 5 conveyances. Using the 5 conveyances, we need to make 3 journeys.
Let us consider the 1st journey.
For the 1st journey, we can use any of the five conveyances. So, the number of ways of making the 1st journey is 5.
Now consider the 2nd journey.
For the 2nd journey, we can use any of the five conveyances. So, the number of ways of making the 2nd journey is also 5.
Now consider the 3rd journey.
For the 3rd journey also, we can use any of the five conveyances. So, the number of ways of making the 3rd journey is also 5.
Now to find the number of ways of making 3 journeys, by taking the product of the number of ways of making 1st, 2nd, and 3rd journeys. It is given by,
 \[5 \times 5 \times 5 = {5^3}\]
Therefore, the number of ways of making 3 journeys using 5 conveyances is ${5^3}$
So, the correct answer is option C.

Note: Alternate solution to this problem is by,
We can compare this problem as selecting 3 objects from 5 objects with replacement. So, by permutations, the number of ways of selecting r objects from n objects with replacement is ${n^r}$ .
Here $n = 5$ and $r = 3$
 $ \Rightarrow {n^r} = {5^3}$
Therefore, the number of ways of making 3 journeys using 5 conveyances is ${5^3}$